In a large saucepot, melt the butter over low heat.
When melted, add the marshmallows. Stir until melted, approximately 3 minutes. Add 1/2 teaspoon peppermint extract and stir to combine.
Add the rice krispies and 2 of the crushed candy canes. Stir to combine.
Immediately transfer the rice krispie mixture to a greased 9 by 13 baking dish. Press the mixture into the dish until level and spread out.
Melt the white chocolate by placing in a bowl and microwaving on high for 1 minute. Stir and continue heating if needed in 15 second intervals, stirring in between, until fully melted. Stir in the remaining 1/4 teaspoon peppermint extract.
Spread the white chocolate across the rice krispies. This is easiest with an offset spatula. Try to make it as even as possible.
Let the rice krispies sit until the white chocolate is firm, approximately 20 minutes in the refrigerator. (This can also be done at room temperature but it will take a lot longer).
Once firm, heat the chocolate chips in a small bowl in the microwave by heating on high for 1 minute. Stir and continue heating if needed in 15 second intervals, stirring in between, until fully melted.
Spread the chocolate across the rice krispies with the offset spatula, again trying to make it as level as possible.
Immediately sprinkle the top with the remaining crushed candy canes and any sprinkles as desired. Cool or let sit until firm. Cut into 12 squares.